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\title{Discovery through Gossip\footnote{A preliminary version of this
    paper appeared in Proceedings of the {\em ACM Symposium on
      Parallelism in Algorithms and Architectures (SPAA)}, 2012, 140-149.}}

\author{Bernhard Haeupler \thanks{Computer Science Department, Carnegie Mellon University, Pittsburgh, PA 15213, USA. E-mail: {\tt haeupler@cs.cmu.edu}. Supported in part by US NSF grant CCF-1527110.}
\and Gopal Pandurangan \thanks{Department of Computer Science, University of Houston, Houston, TX 77204, USA. Work done while at the Division of Mathematical
Sciences, Nanyang Technological University, Singapore 637371 and Department of Computer Science, Brown University, Providence, RI 02912, USA.  \hbox{E-mail}:~{\tt gopalpandurangan@gmail.com}. Supported in part by the following grants: Nanyang Technological University grant M58110000, Singapore Ministry of Education (MOE) Academic Research Fund (AcRF) Tier 2 grant MOE2010-T2-2-082,
US NSF grant CCF-1023166, and  the US-Israel Binational Science
Foundation (BSF) grant 2008348.} \and David Peleg \thanks{Department of Computer Science and Applied Mathematics, The Weizmann
Institute of Science, Rehovot, 76100 Israel.
E-mail: {\tt david.peleg@weizmann.ac.il}.
Supported by a grant from the United States-Israel Binational Science
Foundation (BSF).}
  \and Rajmohan Rajaraman \thanks{College of Computer and Information Science,
  Northeastern University, Boston  MA 02115, USA.
E-mail: {\tt \{rraj,austin\}@ccs.neu.edu}.  Supported in part by NSF grant CNS-0915985.} \and  Zhifeng Sun~$^\S$}

\date{}
 
\maketitle

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\begin{abstract}
  We study randomized gossip-based processes in dynamic networks that are motivated
  by information discovery in large-scale distributed networks such as
  peer-to-peer and social networks.  A well-studied problem in
  peer-to-peer networks is {\em resource discovery}, where the goal
  for nodes (hosts with IP addresses) is to discover the IP addresses
  of all other hosts.  Also, some of the recent work on self-stabilization algorithms for P2P/overlay networks  proceed
  via discovery of the complete network.  In social networks, nodes (people) discover new
  nodes through exchanging contacts with their neighbors (friends).
  In both cases the discovery of new nodes changes the underlying
  network --- new edges are added to the network --- and the process
  continues in the changed network.   Rigorously analyzing such  dynamic (stochastic) processes in a continuously changing topology  remains a challenging problem with obvious applications. 
  
  This paper studies and analyzes two natural gossip-based discovery
  processes.   In the {\em push discovery} or {\em triangulation}\/
  process, each node repeatedly chooses two random neighbors and
  connects them (i.e., ``pushes'' their mutual information to each
  other). In the {\em pull discovery}\/ process or the {\em two-hop
    walk}, each node repeatedly requests or ``pulls'' a random contact
  from a random neighbor and connects itself to this two-hop neighbor.
  Both processes are lightweight in the sense that the amortized work
  done per node is constant per round, local, and naturally robust due
  to the inherent randomized nature of gossip. 
 
  Our main result is an almost-tight analysis of the time taken for
  these two randomized processes to converge.  We show that in any
  undirected $n$-node graph both processes take $O(n\log^2 n)$ rounds
  to connect every node to all other nodes with high probability,
  whereas $\Omega(n \log n)$ is a lower bound.  We also study the
  two-hop walk in directed graphs, and show that it takes $O(n^2 \log
  n)$ time with high probability, and that the worst-case bound is
  tight for arbitrary directed graphs, whereas $\Omega(n^2)$ is a
  lower bound for strongly connected directed graphs.  A key technical
  challenge that we overcome in our work is the analysis of a
  randomized process that itself results in a constantly changing
  network leading to complicated dependencies in every round.  We
  discuss implications of our results and their analysis to discovery
  problems in P2P networks as well as to evolution in social networks.
  %On a broader level, our work also raises interesting questions on how local gossip-based processes influence emergence of key properties in dynamic networks. 
 \end{abstract}

{\bf Keywords:} Random process, Resource discovery, Social network, Gossip-based algorithm,
Distributed algorithm, Probabilistic analysis

 
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\input{2hop-10p}
\input{directed-10p}
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\end{thebibliography}
\fi

\end{document}


\appendix

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\input{prerequisite}
\input{triangulation}
\input{2hop}
\input{directed}
%\input{discussion}

\end{document}
